TF SPIM HadronicInteractions

Proton Bremsstrahlung

[math]\sigma_{class}|_{90^o} = 2.1\times 10^{-31} cm^2/sterad[/math] = cross section for dipole radiation emitted at 90 degrees with respect to incident beam of particles scattered in a Coulomb field.

File:ProtonBrem Drell Huang PhysRev v99 n3 1955 pg686.pdf

Pluto event generator

A ROOT based Hadronic Simulation package based on Pluto

I installed Pluto V 5.14.1 on inca

I needed to set the environmental variables under tcsh

setenv ROOTSYS ~/src/ROOT/root
setenv PATH ${PATH}:${ROOTSYS}/bin
setenv LD_LIBRARY_PATH $ROOTSYS/lib

There is a subdirectory called "macros"

cd macros

Go to that subdirectory and type root, this will run the contents of the file "rootlogin.C"

cd macros

inca:~/src/Pluto/pluto_v5.14.1/macros> root
 *******************************************
 *                                         *
 *        W E L C O M E  to  R O O T       *
 *                                         *
 *   Version   5.17/03    30 August 2007   *
 *                                         *
 *  You are welcome to visit our Web site  *
 *          http://root.cern.ch            *
 *                                         *
 *******************************************
Compiled on 5 September 2007 for linux with thread support.
CINT/ROOT C/C++ Interpreter version 5.16.24, July 26, 2007
Type ? for help. Commands must be C++ statements.
Enclose multiple statements between { }.
 *********************************************************
 * The Pluto event generator                              
 * (C) HADES collaboration and all contributing AUTHORS   
 * www-hades.gsi.de/computing/pluto/html/PlutoIndex.html  
 * Version: 5.14.1
 * Compiled on 10 December 2008
 *********************************************************
Shared library Pluto.so loaded

to run a pp elastic model type

root [0] .x pp_elastic.C 

a root ntuple is generate called "pp_elastic.root"

you can then analyze the data in the root file with

data->MakeClass();

the above command within root generates an analysis skeleton program.

using t.Show you can see the structure of the events within the ntuple. A few functions are also stored in the root tree which you can use. You can use the root file event to create an input file which GEANT4 can then use as its event generator. GEANT4 then reads the events in and propagates them through your geometry.

Neutron Interactions

Name	Energy
Cold Neutron	micro eV
Thermal	[math]\sim \frac{1}{40}[/math] eV
epithermal	[math] \frac{1}{40} eV \rightarrow 100 keV[/math]
fast	[math]100 keV \rightarrow 100 MeV[/math]
high energy	[math] \gt 100 MeV[/math]

Note: Interaction length for neutrons is ~[math]10^{-13}[/math] .
Neutrons are even better than photons for penetrating matter.

Elastic scattering

File:Elastic scattering from Nuclei.jpg

[math]v_{CM} = \frac{m_n v_L + M(0)}{m_n + M} = \frac{v_0}{1 + \frac{M}{m_n}} = \frac{v_0}{1+A} =[/math] velocity of CM frame

[math]{v_L}^' = [/math] Magnitude of Neutron velocity in CM frame before and after collision
[math]= v_c - v_{CM} = v_0 -\frac{v_0}{1+A} = \frac{(1+A)-1}{1+A} v_0 = \frac{A}{1+A} v_0[/math]

[math] v = [/math] Magnitude of Nucleus velocity in CM frame before and after collision
[math]= v_{CM} = \frac{v_0}{1+A}[/math]

Note: In elastic collision only the particles direction changes.

[math]\vec{v}_L = {\vec{v}_L}^' + \vec{v}_{CM}[/math]

File:Rule of cosines.jpg

[math]c^2 = a^2 + b^2 - 2abcos \theta[/math]

[math](v_L)^2 = ({v_L}^')^2 + (v_{CM})^2 - 2 v_{CM} {v_L}^' cos(\pi - {\theta}_{CM})=[/math]

[math]= ({v_L}^')^2 + (V)^2 - 2 V {v_L}^' cos(\pi - {\theta}_{CM})=[/math]

where

[math]({v_L}^')^2 = (\frac{A}{1+A})^2 {v_0}^2[/math]

[math](V)^2 = (\frac{1}{1+A})^2 {v_0}^2[/math]

After substitution we get following:

[math](\frac{v_L}{v_0})^2 = \frac{A^2 +1 - 2 A cos(\pi - {\theta}_{CM})}{(1+A)^2} = \frac{A^2 +1 + 2 A cos({\theta}_{CM})}{(1+A)^2}[/math]

[math] cos(A+/-B) = cosAcosB -/+ sinAsinB[/math]

[math]\frac{E}{E_0} = \frac{A^2 + 1 + 2Acos({\theta}_{CM})}{(1+A)^2}[/math]

when [math]{\theta}_{CM}=0[/math], [math]E_{max} = E_0[/math].

[math]E_{min} = \frac{(A-1)^2}{(A+1)^2} E_0 = (\frac{A-1}{A+1}) E_0 =[/math] Minimum energy of scattered Neutron in LAB frame.

File:Rule of cosines 1.jpg

[math]({v_L}^')^2 = (v_L)^2 + (v_{CM})^2 - 2 v_{CM} v_L cos(\pi - {\theta}_{CM})=[/math]

[math]= (v_L)^2 + (V)^2 - 2 V v_L cos({\theta}_{L})=[/math]

[math](\frac{Av_0}{1+A})^2 = {v_L}^2 + (\frac{v_0}{1+A})^2 - 2 v_L (\frac{v_0}{1+A}) cos({\theta}_L)[/math]

After substituting [math]v_L[/math]

[math]cos{\theta}_L = [\frac{A^2 + 1 + 2 A cos{\theta}_{CM}}{(1+A)^2} + (\frac{1}{1+A})^2 - (\frac{A}{1+A})^2] \times \frac{(1+A)^2}{2\sqrt{A^2 +1 + 2Acos{\theta}_{CM}}} = [/math]

[math]= \frac{[A^2 +1 + 2Acos{\theta}_{CM} + 1 - A^2]}{2\sqrt{A^2 +1 + 2Acos{\theta}_{CM}}} = \frac{1 + Acos{\theta}_{CM}}{\sqrt{A^2 +1 + 2Acos{\theta}_{CM}}} [/math]

Note: [math] {E_A}^{CM} = \frac{1}{2} M_A V^2 = \frac{1}{2} A m_n (\frac{v_0}{1+A})^2 = \frac{A}{(1+A)^2} \frac{m_n {v_0}^2}{2}= [/math]

[math] = \frac{A}{(1+A)^2}E_0 = [/math] Energy of recoil Nuclei in CM frame.

Conservation of Energy: [math]E_0 = E + E_A[/math]

[math] E_A = E_0 - E = E_0 - \frac{A^2 + 1 + 2Acos{\theta}_{CM}}{(1+A)^2} E_0 = [/math]

[math]\frac{(1+A)^2 - (A^2 +1 + 2Acos{\theta}_{CM})}{(1+A)^2}E_0 =[/math]

Lethargy

Lethargy[math] \equiv[/math] u[math] \equiv \ln \frac{E_0}{E}[/math] = logarithmic energy change

=[math]\ln \left ( \frac{(1+A)^2} {A^2 + 1 + 2Acos{\theta}_{CM}} \right )[/math]

The average lethargy

[math]\lt u(\theta)\gt = \int u(\theta) \frac{d \Omega}{4 \pi}[/math]

=[math]\frac{1}{2} \int \ln \left ( \frac{(1+A)^2} {A^2 + 1 + 2Acos{\theta}_{CM}} \right ) d[cos(\theta)][/math]

N = # of collisions = [math]\frac{\mbox{Desired Energy Change}}{\mbox{Average Energy change per collision}}[/math]

[math]= \frac{\ln \left (\frac{E_0}{E} \right )}{1}[/math]

Inelastic Scattering

Simulations_of_Particle_Interactions_with_Matter